Don't organists give of their time freely, or as good as, to the likes of churches? Many do/did in the UK, from what I can remember.
A brief search didn't turn up much in the way of web resources for organists in France. At least, not sites that had been updated recently.
(Going off-topic, it's funny what can turn up in search results. This article from the Lyon-based Progrès is about an organist who was eventually tracked down for his crimes - a serial killer of local cats, 17 in total! He got three months suspended. It doesn't say if he lost his job...)
Good question - how exactly do professional organists support themselves in France? Frankly, I can't think of a church I've been in that even has an organ... though I must say, I've not one to frequent churches.
I suspect a professional musician would either be considered a "performance artist" and do concerts, or would give lessons, possibly through a music school of some kind.
In France, a professional organist would make a living in just the same way as in the US:
Salaried organist position in a cathedral or a large church (Notre Dame de Paris, St-Sulpice, La Madeleine, Sacré-Coeur etc)
With major orchestra
On the staff of music college (conservatoire), university, music teacher in school
and for a select few, as concert musician.
Many organists combine two or more roles.
As well as finding a job, there is a tricky question of work permit and visa. If you don't have a passport from an EU country, you need a work visa, and none will be issued unless a) you are a musician of international repute and can get an artist visa or your employer will go through a work permit process for you; or b) you get one for another, unrelated job or as student/researcher (e.g. PhD) and work in your spare time by playing in churches or giving private lessons.
Also you need, for most jobs, a high degree of French, and it's virtually impossible to get a position as teacher in state schools as they normally only appoint French nationals (not even other EU citizens) as civil servants who are directly employed by Ministry of National Education.